RS Aggarwal Solutions: Exercise 4D - Quadratic Equations Class 10 Notes | EduRev

Mathematics (Maths) Class 10

Class 10 : RS Aggarwal Solutions: Exercise 4D - Quadratic Equations Class 10 Notes | EduRev

 Page 1


1. Find the nature of roots of the following quadratic equations:
(i) 
2
2 8 5 0. x x
(ii) 
2
3 2 6 2 0. x x
(iii) 
2
5 4 1 0. x x
(iv) 5 2 6 0 x x
(v)
2
12 4 15 5 0 x x
(vi) 
2
2 0. x x
Sol:
(i) The given equation is
2
2 8 5 0. x x
This is of the form
2
0, a x b x c where 2, 8 5. a b a n d c
Discriminant, 
2
2
4 8 4 2 5 64 40 24 0 D b a c
Hence, the given equation has real and unequal roots.
(ii) The given equation is
2
3 2 6 2 0. x x
This is of the form
2
0, a x b x c where 3, 2 6 2. a b a n d c
Discriminant, 
2
2
4 2 6 4 3 2 24 24 0 D b a c
Hence, the given equation has real and equal roots.
(iii) The given equation is
2
5 4 1 0. x x
This is of the form
2
0, a x b x c where 5, 4 1. a b a n d c
Discriminant, 
2
2
4 4 4 5 1 16 20 4 0 D b a c
Hence, the given equation has no real roots.
(iv) The given equation is
Page 2


1. Find the nature of roots of the following quadratic equations:
(i) 
2
2 8 5 0. x x
(ii) 
2
3 2 6 2 0. x x
(iii) 
2
5 4 1 0. x x
(iv) 5 2 6 0 x x
(v)
2
12 4 15 5 0 x x
(vi) 
2
2 0. x x
Sol:
(i) The given equation is
2
2 8 5 0. x x
This is of the form
2
0, a x b x c where 2, 8 5. a b a n d c
Discriminant, 
2
2
4 8 4 2 5 64 40 24 0 D b a c
Hence, the given equation has real and unequal roots.
(ii) The given equation is
2
3 2 6 2 0. x x
This is of the form
2
0, a x b x c where 3, 2 6 2. a b a n d c
Discriminant, 
2
2
4 2 6 4 3 2 24 24 0 D b a c
Hence, the given equation has real and equal roots.
(iii) The given equation is
2
5 4 1 0. x x
This is of the form
2
0, a x b x c where 5, 4 1. a b a n d c
Discriminant, 
2
2
4 4 4 5 1 16 20 4 0 D b a c
Hence, the given equation has no real roots.
(iv) The given equation is
 
 
2
5 2 6 0
5 10 6 0
x x
x x
This is of the form 
2
0, a x b x c where 5, 10 6. a b a n d c
Discriminant, 
2
2
4 10 4 5 6 100 120 20 0 D b a c
Hence, the given equation has no real roots.
(v) The given equation is 
2
12 4 15 5 0 x x
This is of the form 
2
0, a x b x c where 12, 4 15 5. a b a n d c
Discriminant, 
2
2
4 4 15 4 12 5 240 240 0 D b a c
Hence, the given equation has real and equal roots.
(vi) The given equation is 
2
2 0. x x
This is of the form 
2
0, a x b x c where 1, 1 2. a b a n d c
Discriminant, 
2
2
4 1 4 1 2 1 8 7 0 D b a c
Hence, the given equation has no real roots.
2. If a and b are distinct real numbers, show that the quadratic equations
2 2 2
2 2 1 0. a b x a b x has no real roots.
Sol:
The given equation is 
2 2 2
2 2 1 0. a b x a b x
2
2 2
2 2 2 2
2 2 2 2
2 2
2 2
2
2 4 2 1
4 2 8
4 8 4 8 8
4 8 4
4 2
4 0
D a b a b
a a b b a b
a a b b a b
a a b b
a a b b
a b
Hence, the given equation has no real roots.
3. Show that the roots of the equation 
2 2
0 x p x q are real for all real values of p and q.
Sol:
Given:
2 2
0 x p x q
Here,
2
1, a b p a n d c q
Page 3


1. Find the nature of roots of the following quadratic equations:
(i) 
2
2 8 5 0. x x
(ii) 
2
3 2 6 2 0. x x
(iii) 
2
5 4 1 0. x x
(iv) 5 2 6 0 x x
(v)
2
12 4 15 5 0 x x
(vi) 
2
2 0. x x
Sol:
(i) The given equation is
2
2 8 5 0. x x
This is of the form
2
0, a x b x c where 2, 8 5. a b a n d c
Discriminant, 
2
2
4 8 4 2 5 64 40 24 0 D b a c
Hence, the given equation has real and unequal roots.
(ii) The given equation is
2
3 2 6 2 0. x x
This is of the form
2
0, a x b x c where 3, 2 6 2. a b a n d c
Discriminant, 
2
2
4 2 6 4 3 2 24 24 0 D b a c
Hence, the given equation has real and equal roots.
(iii) The given equation is
2
5 4 1 0. x x
This is of the form
2
0, a x b x c where 5, 4 1. a b a n d c
Discriminant, 
2
2
4 4 4 5 1 16 20 4 0 D b a c
Hence, the given equation has no real roots.
(iv) The given equation is
 
 
2
5 2 6 0
5 10 6 0
x x
x x
This is of the form 
2
0, a x b x c where 5, 10 6. a b a n d c
Discriminant, 
2
2
4 10 4 5 6 100 120 20 0 D b a c
Hence, the given equation has no real roots.
(v) The given equation is 
2
12 4 15 5 0 x x
This is of the form 
2
0, a x b x c where 12, 4 15 5. a b a n d c
Discriminant, 
2
2
4 4 15 4 12 5 240 240 0 D b a c
Hence, the given equation has real and equal roots.
(vi) The given equation is 
2
2 0. x x
This is of the form 
2
0, a x b x c where 1, 1 2. a b a n d c
Discriminant, 
2
2
4 1 4 1 2 1 8 7 0 D b a c
Hence, the given equation has no real roots.
2. If a and b are distinct real numbers, show that the quadratic equations
2 2 2
2 2 1 0. a b x a b x has no real roots.
Sol:
The given equation is 
2 2 2
2 2 1 0. a b x a b x
2
2 2
2 2 2 2
2 2 2 2
2 2
2 2
2
2 4 2 1
4 2 8
4 8 4 8 8
4 8 4
4 2
4 0
D a b a b
a a b b a b
a a b b a b
a a b b
a a b b
a b
Hence, the given equation has no real roots.
3. Show that the roots of the equation 
2 2
0 x p x q are real for all real values of p and q.
Sol:
Given:
2 2
0 x p x q
Here,
2
1, a b p a n d c q
 
 
Discriminant D is given by:
2
2 2
2 2
4
4 1
4 0
D b a c
p q
p q
0 D for all real values of p and q.
Thus, the roots of the equation are real.
4. For what values of k are the roots of the quadratic equation 
2
3 2 27 0 x k x real and 
equal?
Sol:
Given:
2
3 2 27 0 x k x
Here,
3, 2 27 a b k a n d c
It is given that the roots of the equation are real and equal; therefore, we have:
2
2
0
2 4 3 27 0
4 324 0
D
k
k
2
2
4 324
81
9
9 9
k
k
k
k o r k
5. For what value of k are the roots of the quadratic equation 2 5 10 0 k x x real and 
equal.
Sol:
The given equation is
2
2 5 10 0
2 5 10 0
k x x
k x k x
This is of the form 
2
0, a x b x c where , 2 5 10. a k b k a n d c
2
2 2
4 2 5 4 10 20 40 D b a c k k k k
The given equation will have real and equal roots if 0. D
Page 4


1. Find the nature of roots of the following quadratic equations:
(i) 
2
2 8 5 0. x x
(ii) 
2
3 2 6 2 0. x x
(iii) 
2
5 4 1 0. x x
(iv) 5 2 6 0 x x
(v)
2
12 4 15 5 0 x x
(vi) 
2
2 0. x x
Sol:
(i) The given equation is
2
2 8 5 0. x x
This is of the form
2
0, a x b x c where 2, 8 5. a b a n d c
Discriminant, 
2
2
4 8 4 2 5 64 40 24 0 D b a c
Hence, the given equation has real and unequal roots.
(ii) The given equation is
2
3 2 6 2 0. x x
This is of the form
2
0, a x b x c where 3, 2 6 2. a b a n d c
Discriminant, 
2
2
4 2 6 4 3 2 24 24 0 D b a c
Hence, the given equation has real and equal roots.
(iii) The given equation is
2
5 4 1 0. x x
This is of the form
2
0, a x b x c where 5, 4 1. a b a n d c
Discriminant, 
2
2
4 4 4 5 1 16 20 4 0 D b a c
Hence, the given equation has no real roots.
(iv) The given equation is
 
 
2
5 2 6 0
5 10 6 0
x x
x x
This is of the form 
2
0, a x b x c where 5, 10 6. a b a n d c
Discriminant, 
2
2
4 10 4 5 6 100 120 20 0 D b a c
Hence, the given equation has no real roots.
(v) The given equation is 
2
12 4 15 5 0 x x
This is of the form 
2
0, a x b x c where 12, 4 15 5. a b a n d c
Discriminant, 
2
2
4 4 15 4 12 5 240 240 0 D b a c
Hence, the given equation has real and equal roots.
(vi) The given equation is 
2
2 0. x x
This is of the form 
2
0, a x b x c where 1, 1 2. a b a n d c
Discriminant, 
2
2
4 1 4 1 2 1 8 7 0 D b a c
Hence, the given equation has no real roots.
2. If a and b are distinct real numbers, show that the quadratic equations
2 2 2
2 2 1 0. a b x a b x has no real roots.
Sol:
The given equation is 
2 2 2
2 2 1 0. a b x a b x
2
2 2
2 2 2 2
2 2 2 2
2 2
2 2
2
2 4 2 1
4 2 8
4 8 4 8 8
4 8 4
4 2
4 0
D a b a b
a a b b a b
a a b b a b
a a b b
a a b b
a b
Hence, the given equation has no real roots.
3. Show that the roots of the equation 
2 2
0 x p x q are real for all real values of p and q.
Sol:
Given:
2 2
0 x p x q
Here,
2
1, a b p a n d c q
 
 
Discriminant D is given by:
2
2 2
2 2
4
4 1
4 0
D b a c
p q
p q
0 D for all real values of p and q.
Thus, the roots of the equation are real.
4. For what values of k are the roots of the quadratic equation 
2
3 2 27 0 x k x real and 
equal?
Sol:
Given:
2
3 2 27 0 x k x
Here,
3, 2 27 a b k a n d c
It is given that the roots of the equation are real and equal; therefore, we have:
2
2
0
2 4 3 27 0
4 324 0
D
k
k
2
2
4 324
81
9
9 9
k
k
k
k o r k
5. For what value of k are the roots of the quadratic equation 2 5 10 0 k x x real and 
equal.
Sol:
The given equation is
2
2 5 10 0
2 5 10 0
k x x
k x k x
This is of the form 
2
0, a x b x c where , 2 5 10. a k b k a n d c
2
2 2
4 2 5 4 10 20 40 D b a c k k k k
The given equation will have real and equal roots if 0. D
 
 
2
20 40 0
20 2 0
0 2 0
0 2
k k
k k
k o r k
k o r k
But, for 0, k we get 10 0, which is not true
Hence, 2 is the required value of k.
6. For what values of p are the  roots of the equation 
2
4 3 0. x p x real and equal?
Sol:
The given equation is 
2
4 3 0. x p x
This is of the form 
2
0, a x b x c where 4, 3. a b p a n d c
2 2 2
4 4 4 3 48 D b a c p p
The given equation will have real and equal roots if 0. D
2
2
48 0
48
48 4 3
p
p
p
Hence, 4 3 and 4 3 are the required values of p.
7. Find the nonzero value of k for which the roots of the quadratic equation 
2
9 3 0. x k x k
are real and equal.
Sol:
The given equation is 
2
9 3 0. x k x k
This is of the form 
2
0, a x b x c where 9, 3 . a b k a n d c k
2
2 2
4 3 4 9 9 36 D b a c k k k k
The given equation will have real and equal roots if 0. D
2
9 36 0
9 4 0
0 4 0
0 4
k k
k k
k o r k
k o r k
But, 0 k (Given)
Hence, the required values of k is 4.
8. Find the values of k for which the quadratic equation 
2
3 1 2 1 1 0. k x k x has real 
and equal roots.
Sol:
Page 5


1. Find the nature of roots of the following quadratic equations:
(i) 
2
2 8 5 0. x x
(ii) 
2
3 2 6 2 0. x x
(iii) 
2
5 4 1 0. x x
(iv) 5 2 6 0 x x
(v)
2
12 4 15 5 0 x x
(vi) 
2
2 0. x x
Sol:
(i) The given equation is
2
2 8 5 0. x x
This is of the form
2
0, a x b x c where 2, 8 5. a b a n d c
Discriminant, 
2
2
4 8 4 2 5 64 40 24 0 D b a c
Hence, the given equation has real and unequal roots.
(ii) The given equation is
2
3 2 6 2 0. x x
This is of the form
2
0, a x b x c where 3, 2 6 2. a b a n d c
Discriminant, 
2
2
4 2 6 4 3 2 24 24 0 D b a c
Hence, the given equation has real and equal roots.
(iii) The given equation is
2
5 4 1 0. x x
This is of the form
2
0, a x b x c where 5, 4 1. a b a n d c
Discriminant, 
2
2
4 4 4 5 1 16 20 4 0 D b a c
Hence, the given equation has no real roots.
(iv) The given equation is
 
 
2
5 2 6 0
5 10 6 0
x x
x x
This is of the form 
2
0, a x b x c where 5, 10 6. a b a n d c
Discriminant, 
2
2
4 10 4 5 6 100 120 20 0 D b a c
Hence, the given equation has no real roots.
(v) The given equation is 
2
12 4 15 5 0 x x
This is of the form 
2
0, a x b x c where 12, 4 15 5. a b a n d c
Discriminant, 
2
2
4 4 15 4 12 5 240 240 0 D b a c
Hence, the given equation has real and equal roots.
(vi) The given equation is 
2
2 0. x x
This is of the form 
2
0, a x b x c where 1, 1 2. a b a n d c
Discriminant, 
2
2
4 1 4 1 2 1 8 7 0 D b a c
Hence, the given equation has no real roots.
2. If a and b are distinct real numbers, show that the quadratic equations
2 2 2
2 2 1 0. a b x a b x has no real roots.
Sol:
The given equation is 
2 2 2
2 2 1 0. a b x a b x
2
2 2
2 2 2 2
2 2 2 2
2 2
2 2
2
2 4 2 1
4 2 8
4 8 4 8 8
4 8 4
4 2
4 0
D a b a b
a a b b a b
a a b b a b
a a b b
a a b b
a b
Hence, the given equation has no real roots.
3. Show that the roots of the equation 
2 2
0 x p x q are real for all real values of p and q.
Sol:
Given:
2 2
0 x p x q
Here,
2
1, a b p a n d c q
 
 
Discriminant D is given by:
2
2 2
2 2
4
4 1
4 0
D b a c
p q
p q
0 D for all real values of p and q.
Thus, the roots of the equation are real.
4. For what values of k are the roots of the quadratic equation 
2
3 2 27 0 x k x real and 
equal?
Sol:
Given:
2
3 2 27 0 x k x
Here,
3, 2 27 a b k a n d c
It is given that the roots of the equation are real and equal; therefore, we have:
2
2
0
2 4 3 27 0
4 324 0
D
k
k
2
2
4 324
81
9
9 9
k
k
k
k o r k
5. For what value of k are the roots of the quadratic equation 2 5 10 0 k x x real and 
equal.
Sol:
The given equation is
2
2 5 10 0
2 5 10 0
k x x
k x k x
This is of the form 
2
0, a x b x c where , 2 5 10. a k b k a n d c
2
2 2
4 2 5 4 10 20 40 D b a c k k k k
The given equation will have real and equal roots if 0. D
 
 
2
20 40 0
20 2 0
0 2 0
0 2
k k
k k
k o r k
k o r k
But, for 0, k we get 10 0, which is not true
Hence, 2 is the required value of k.
6. For what values of p are the  roots of the equation 
2
4 3 0. x p x real and equal?
Sol:
The given equation is 
2
4 3 0. x p x
This is of the form 
2
0, a x b x c where 4, 3. a b p a n d c
2 2 2
4 4 4 3 48 D b a c p p
The given equation will have real and equal roots if 0. D
2
2
48 0
48
48 4 3
p
p
p
Hence, 4 3 and 4 3 are the required values of p.
7. Find the nonzero value of k for which the roots of the quadratic equation 
2
9 3 0. x k x k
are real and equal.
Sol:
The given equation is 
2
9 3 0. x k x k
This is of the form 
2
0, a x b x c where 9, 3 . a b k a n d c k
2
2 2
4 3 4 9 9 36 D b a c k k k k
The given equation will have real and equal roots if 0. D
2
9 36 0
9 4 0
0 4 0
0 4
k k
k k
k o r k
k o r k
But, 0 k (Given)
Hence, the required values of k is 4.
8. Find the values of k for which the quadratic equation 
2
3 1 2 1 1 0. k x k x has real 
and equal roots.
Sol:
 
 
The given equation is 
2
3 1 2 1 1 0. k x k x
This is of the form 
2
0, a x b x c where 3 1, 2 1 1. a k b k a n d c
2
2
2
2
2
4
2 1 4 3 1 1
4 2 1 4 3 1
4 8 4 12 4
4 4
D b ac
k k
k k k
k k k
k k
The given equation will have real and equal roots if 0. D
2
4 4 0
4 1 0
0 1 0
0 1
k k
k k
k o r k
k o r k
Hence, 0 and 1are the required values of k.
9. Find the value of p for which the quadratic equation 
2
2 1 7 2 7 3 0. p x p x p
has real and equal roots.
Sol:
The given equation is 
2
2 1 7 2 7 3 0. p x p x p
This is of the form 
2
0, a x b x c where 2 1, 7 2 7 3. a p b p a n d c p
2
4 D b a c
2
7 2 4 2 1 7 3 p p p
2 2
49 28 4 4 14 3 p p p p
2 2
49 28 4 56 4 12 p p p p
2
7 24 16 p p
The given equation will have real and equal roots if 0. D
2
7 24 16 0 p p
2
7 24 16 0 p p
2
7 28 4 16 0 p p p
7 4 4 4 0 p p p
4 7 4 0 p p
4 0 7 4 0 p o r p
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